关于Horn问题及其体积函数
On Horn's Problem and its Volume Function
摘要：
本文考虑Horn问题的一个推广：给定紧李群线性表示的两个轨道$\mathcal{O}_α$和$\mathcal{O}_β$，设$A\in \mathcal{O}_α$，$B\in \mathcal{O}_β$是这两个轨道上独立的不变分布随机元.问题是描述和$A+B$的轨道的概率分布。我们特别研究了常见的余共轭轨道的情形，以及自共轭实矩阵、复矩阵和四元数矩阵分别在$\mathrm{SO}(n)$、$\mathrm{SU}(n)$和$\mathrm{USp}(n)$共轭作用下的轨道.概率密度可以用一个函数来表示，我们称之为体积函数。在本文中，（i）我们将此功能的辛或黎曼几何的轨道，根据情况而定;（ii）我们讨论其非解析性和可能的消失;（iii）在coadjoint的情况下，我们研究它的关系张量积多重性（广义Littlewood-理查森系数），并表明它计算家庭的凸多面体引入贝伦斯坦和Zelevinsky的体积。这些考虑说明了一个详细的研究体积函数的coadjoint轨道的$B_2=\mathfrak{so}(5)$。
英文摘要：
We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_α$ and $\mathcal{O}_β$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_α$, $B\in \mathcal{O}_β$ be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum $A+B$. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm{USp}(n)$ respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood--Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of $B_2=\mathfrak{so}(5)$.
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